Linearity and the Mathematics of Several Variables

The following sections are included:

• Contents

• Preface

The following sections are included:

• Vectors that You Know
  • Exercises

• Lines and Planes
  • Exercises

• Points: A Deeper Look
  • Exercises

• Curves and Tangent Vectors
  • Exercises

The following sections are included:

• Linear Systems and Matrices
  • Solving linear equations
  • More examples and variations
  • Exercises

• Matrix Algebra
  • Economic applications
  • Additional examples
  • Exercises

• Inverses
  • Facts and remarks about inverses
  • Finding the inverse by row reduction
  • More examples
  • Exercises

• Functions and Gradient Vectors
  • The chain rule
  • Exercises

• Elementary Determinants
  • Exercises

The following sections are included:

• The Definition of a Vector Space
  • Examples
  • Exercises

• Linear Functions
  • Domain and codomain
  • Linear functions from Rⁿ into R^p
  • Affine functions
  • Geometry and physics
  • Operations on functions (Building new linear functions from old)
  • Exercises

• Nonlinear Functions
  • Limits and continuity
  • Exercises

• Differentials
  • A better example
  • Applications and examples
  • Exercises

• The Chain Rule
  • Exercises

The following sections are included:

• The Basis Concept: Independence and Span
  • Span
  • Linear independence
  • Bases
  • Testing a set for linear independence
  • Exercises

• Local Bases Associated with a Coordinate System
  • Vector-valued functions in curvilinear coordinates (An example)
  • Exercises

• Dimension
  • Summary theorem on dimension, independence, and spanning
  • Lemmas and proofs
  • Exercises

• Coordinates with Respect to a Basis
  • The matrix of a linear function with respect to a basis
  • Examples of finding the matrix of a transformation
  • Leaving or returning to the natural basis
  • Exercises

• Change of Basis
  • General theorem on change of basis
  • More examples
  • Transforming row vectors
  • Exercises

The following sections are included:

• Subspaces
  • Examples
  • Affine subspaces
  • Intersections, unions, and sums
  • Exercises

• Subspaces Associated with a Linear Function: Kernel and Range
  • Range
  • One-to-one and onto
  • Kernel
  • Summary examples
  • Isomorphisms
  • Exercises

• Linear Equations: The Superposition Principles
  • Homogeneous vs. nonhomogeneous equations
  • Solving linear problems
  • Summary of the relations among subspaces, linear functions, and linear equations
  • Exercises

• Rank
  • Application of the theorem to algebraic linear equations
  • More about rank
  • Exercises

• Implicit and Inverse Functions
  • The implicit function theorem
  • Postscript
  • Exercises

The following sections are included:

• Inner Products, Norms, and Metrics
  • Inner products, etc., in general vector spaces
  • Elementary theorems about inner products
  • Exercises

• Orthogonality
  • The Gram-Schmidt process
  • Gram-Schmidt examples
  • Exercises

• The Geometry of Curves
  • Differentiation in abstract vector spaces
  • Tangent vectors, unit tangent vectors, and arc length
  • Normal vectors, binormal vectors, curvature, torsion
  • Line integrals
  • Exercises

• Nabla: The Vector Differential Operations
  • Relations between ∇ and the Jacobian matrix
  • A survey of identities satisfied by ∇
  • Exercises

The following sections are included:

• Properties of Determinants
  • Additional examples
  • Quadrature rules
  • Exercises

• Volume, Rotations, and All That: The Geometrical Significance of Determinants and Antisymmetry
  • Can the cross product be generalized to other dimensions?
  • The cross product unmasked
  • Exercises

• Jacobi's Theorem: Changing Variables in Multiple Integrals
  • General statement of Jacobi's theorem
  • Exercises

• Surface Integrals — Definition
  • Euclidean geometry versus differential forms
  • Exercises

• The Integral Theorems of Vector Analysis
  • Intuitive significance and integral definition of div and curl
  • Indirect methods of evaluating line and surface integrals
  • Path independence and potentials
  • Holes in space
  • Exercises

• Direct Methods of Evaluating Line and Surface Integrals
  • Direct methods I: The algebraic viewpoint
  • Direct methods II: The euclidean geometric viewpoint
  • Exercises

The following sections are included:

• Eigenvalues and Eigenvectors
  • Why eigenvectors are nice
  • How to find the eigenvalues and eigenvectors (When they exist)
  • Can it be done?
  • Examples
  • Exercises

• Diagonalization of Real, Symmetric Matrices
  • Quadratic forms
  • Exercises

• A Note on History

The following sections are included:

• Index

• The Greek Alphabet